White-space around counter values

You have several unprotected endlines, but also redundant {...} groups. And the code can be streamlined.

\documentclass{article}
\newcounter{hour}
\newcounter{minute}

\newcommand{\Time}[2]{%
  \setcounter{hour}{#1}%
  \setcounter{minute}{#2}%
}
\newcommand*{\Next}[1][20]{%
  \addtocounter{minute}{#1}%
  \ifnum\value{minute}>59
    \stepcounter{hour}%
    \addtocounter{minute}{-60}%
  \fi
  \ifnum\value{hour}<10 0\fi\thehour
  :%
  \ifnum\value{minute}<10 0\fi\theminute
}
\begin{document}

\Time{8}{0}

Given time is \thehour:\theminute, properly written as \Next[0].
Next default time slot is at \Next. Next slot in 10 minutes is at \Next[10].

\Next[29] -- \Next[1]

\end{document}

Note \stepcounter rather than \refstepcounter.

However, this won't work with an increment of more than 60 minutes and the hour would be wrong if we go past midnight.

\documentclass{article}

\ExplSyntaxOn
\NewDocumentCommand{\settime}{mm}
 {
  \subhajit_time_set:nn {#1} {#2}
 }
\NewDocumentCommand{\nexttime}{O{20}}
 {
  \subhajit_time_next:n {#1}
 }

\int_new:N \g_subhajit_time_hour_int
\int_new:N \g_subhajit_time_minute_int

\cs_new_protected:Nn \subhajit_time_set:nn
 {
  \int_gset:Nn \g_subhajit_time_hour_int {#1}
  \int_gset:Nn \g_subhajit_time_minute_int {#2}
 }

\cs_new_protected:Nn \subhajit_time_next:n
 {
  \int_gadd:Nn \g_subhajit_time_minute_int {#1}
  % we might go beyond 60
  \int_gset:Nn \g_subhajit_time_hour_int
   {
    \g_subhajit_time_hour_int % current hour
    +
    \int_div_truncate:nn { \g_subhajit_time_minute_int } {60} % plus ...
   }
  \int_gset:Nn \g_subhajit_time_hour_int
   {% maybe we go beyond 24
    \int_mod:nn { \g_subhajit_time_hour_int } {24}
   }
  \int_gset:Nn \g_subhajit_time_minute_int
   {% reduce modulo 60
    \int_mod:nn { \g_subhajit_time_minute_int } {60}
   }
  % print
  \int_compare:nT { \g_subhajit_time_hour_int < 10 } {0}
  \int_to_arabic:n { \g_subhajit_time_hour_int }
  :
  \int_compare:nT { \g_subhajit_time_minute_int < 10 } {0}
  \int_to_arabic:n { \g_subhajit_time_minute_int }
 }
\ExplSyntaxOff

\begin{document}

\settime{8}{0}

\nexttime[0]. 

Next default time slot is at \nexttime. Next slot in 10 minutes is at \nexttime[10].

\nexttime[29] -- \nexttime[1]

\nexttime[654]

\nexttime[654]

\end{document}